## What is/are Bayesian Inversion?

Bayesian Inversion - A semi-analytical Bayesian framework of nonlinear sparse Bayesian learning (NSBL) is proposed to identify sparsity among model parameters during Bayesian inversion.^{[1]}A novel nondestructive method for complete elastic characterization of substrate-coating bilayer specimens with distinct anisotropic layers via resonant ultrasound spectroscopy (RUS) and Bayesian inversion is developed here.

^{[2]}Bayesian inversion of InSAR data from both ascending and descending orbits suggests that the earthquake ruptured a shallow (Depth ~ 5 km), near-horizontal (Dip ~2.

^{[3]}Many parameter estimation problems arising in applications are best cast in the framework of Bayesian inversion.

^{[4]}Finally, I perform a Bayesian inversion of the apparent source function, in order to obtain a kinematic model of the rupture propagation (slip distribution, rupture velocity).

^{[5]}We explore the ability of different approaches to retrieve high frequency converted phases that will be used in the framework of the Bayesian inversion.

^{[6]}The combined approach inherits the merits of the deterministic method and Bayesian inversion as demonstrated by the numerical examples.

^{[7]}The uncertainty and the correlation of inversion parameters were presented using marginal probability distributions and a covariance matrix through Bayesian inversion.

^{[8]}A quality-Bayesian approach, combining the direct sampling method and the Bayesian inversion, is proposed to reconstruct the locations and intensities of the unknown acoustic sources using partial data.

^{[9]}Using the linear flow production data and traditional RTA equations, Bayesian inversion was carried out using two distinct Bayesian methods.

^{[10]}The framework takes the spatially resolved emission inventory TNO-GHGco (1 km x 1 km) as a prior estimate and refines it through the Bayesian inversion of the EM27/SUN observations.

^{[11]}Emissions were calculated by two different top-down methods, a tracer-ratio method (TRM) with carbon monoxide (CO) as the independent tracer, and a Bayesian inversion (BI), based on atmospheric transport simulations using FLEXPART–COSMO.

^{[12]}In contrast, our approach involves a massive data set of short period measurements and a Bayesian inversion that accounts for thin layering.

^{[13]}Bayesian inversion of electromagnetic data produces crucial uncertainty information on inferred subsurface resistivity.

^{[14]}The transfer functions were estimated using an automated algorithm for Bayesian inversion that allows inferring a continuous and objective synchronization between Greenland ice-core and Hulu Cave proxy signals.

^{[15]}BackgroundIn order to use in situ measurements to constrain urban anthropogenic emissions of carbon dioxide (CO2), we use a Lagrangian methodology based on diffusive backward trajectory tracer reconstructions and Bayesian inversion.

^{[16]}Importantly, the Bayesian inversion is carried out by solving a variational optimization problem, replacing traditional computationally-expensive Monte Carlo sampling.

^{[17]}In this work, we utilize such a stochastic prediction and Bayesian inversion and demonstrate results on benchmark problems.

^{[18]}In this work, Bayesian inversion and model selection techniques are applied to compare combinations of three geoacoustic models and corresponding scattering models—the fluid model with the effective density fluid model (EDFM), the grain-shearing elastic model with the viscosity grain-shearing (VGS(λ)) model, and the poroelastic model with the corrected and reparametrized extended Biot–Stoll (CREB) model.

^{[19]}The methodology is composed of three main components: (i) a channel generator to 25 produce networks of the subglacial system; (ii) a physical model that computes pressure and 26 mass transport in steady state; and (iii) Bayesian inversion in which the outputs (pressure, tracer27 transit times) are compared with synthetic data, thus allowing for parameter estimation and 28 uncertainty quantification.

^{[20]}In contrast, the inversion using bias-corrected retrievals from the Greenhouse Gases Observing Satellite (GOSAT) or, to a larger extent, a non-Bayesian inversion that simply adjusts a recent bottom-up flux estimate with the annual growth rate diagnosed from marine surface measurements, estimate much different fluxes and fit the aircraft data less.

^{[21]}The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory.

^{[22]}The numerical implementation of the Bayesian inversion by using a Markov chain Monte Carlo (MCMC) method is computationally challenging because the support of the posterior is restricted to a set of symmetric positive-definite matrices and the dimensionality of the problem grows with the square of the matrix dimension and hence can be high.

^{[23]}In addition, multiple bed boundaries and reservoir images near the borehole are readily obtained by using the Bayesian inversion.

^{[24]}1 of the manuscript to clarify: Atmospheric tomography, a term inspired from medical imaging, combines data from a collection of measurement sites with Bayesian inversion to detect and quantify emissions.

^{[25]}The Bayesian inversion was introduced in the proposed approach for estimating the density function and the Probability of Error metric (PoE) was used to evaluate the tracking accuracy of the system.

^{[26]}These parameters are estimated from a small number of real material images using Bayesian inversion.

^{[27]}

## Transdimensional Bayesian Inversion

For further qualitative interpretation of these findings, we conduct transdimensional Bayesian inversion for S-wave velocity models.^{[1]}The shear-wave velocity and thickness of the sedimentary layers at the investigated slopes are inferred using a transdimensional Bayesian inversion algorithm.

^{[2]}For further qualitative interpretation of these findings, we conduct transdimensional Bayesian inversion for S-wave velocity models.

^{[3]}A recently introduced transdimensional Bayesian inversion procedure is applied for both tomographical methods, which adjusts the fracture positions, orientations, and numbers based on given geometrical fracture statistics.

^{[4]}We use a transdimensional Bayesian inversion scheme constrained by geodetic surface displacements and regularized using von Karman correlation.

^{[5]}

## Dimensional Bayesian Inversion

Here we present an S-wave tomography model at global scale for the Lowermost Mantle (LM) using the Hierarchical Trans-dimensional Bayesian Inversion (HTDBI) framework, LM-HTDBI.^{[1]}The reversible jump Markov Chain Monte Carlo algorithm is a trans-dimensional Bayesian inversion method, which can not only obtain the best inversion solution, but also provide the uncertainty information of inversion parameters, so as to effectively evaluate the reliability of inversion results.

^{[2]}Here we present an S-wave tomography model at global scale for the Lowermost Mantle (LM) using the Hierarchical Trans-dimensional Bayesian Inversion (HTDBI) framework, LM-HTDBI.

^{[3]}The conventional trans-dimensional Bayesian inversion uses Monte Carlo method to search the model space for a solution that satisfies both the acceptance probability and data fitting.

^{[4]}Trans-dimensional Bayesian inversion is applied to the modal dispersion data, based on probabilistic sampling over an unknown number of seabed layers.

^{[5]}

## Nonlinear Bayesian Inversion

We demonstrate how a fully nonlinear Bayesian inversion of surface wave dispersion curves can retrieve the temperature and viscosity fields, without having to explicitly parametrize the elastic tensor.^{[1]}Finally, based on the ingeniously developed nonlinear Bayesian inversion theory, the seafloor shear wave velocity profile in the southern Yellow Sea of China is inverted by employing multi-order Scholte wave dispersion curves.

^{[2]}In this study, we develop an optimal moving window nonlinear Bayesian inversion framework to use the Soil Canopy Observation Photochemistry and Energy fluxes (SCOPE) model for constraining Vcmax , BB slope and LAI with observations of coupled carbon and energy fluxes and spectral reflectance from satellites.

^{[3]}This paper applies nonlinear Bayesian inversion to vector sensor data to estimate seabed geoacoustic properties and uncertainties in South China Sea.

^{[4]}

## Atmospheric Bayesian Inversion

In this study, the atmospheric Bayesian inversion approach that couples six in-situ continuous CO2 monitoring stations with the WRF-Chem transport model at 1-km horizontal resolutions has been used to quantify the impacts of lockdown on CO_{2}emissions for the Paris megacity.

^{[1]}In this study, we use a multi-species atmospheric Bayesian inversion approach to attribute satellite-observed atmospheric CO variations to its sources and sinks in order to achieve a full closure of the global CO budget during 2000–2017.

^{[2]}In this study, we use a multi-species atmospheric Bayesian inversion approach to attribute satellite-observed atmospheric CO variations to its sources and sinks in order to achieve a full closure of the global CO budget during 2000–2017.

^{[3]}

## Use Bayesian Inversion

One approach is to use Bayesian inversion to combine prior assumptions (prior models) with indirect measurements to predict soil parameters and their uncertainty, which can be expressed in form of a posterior parameter distribution.^{[1]}Here, we show how one can use Bayesian inversion to obtain statistical estimates for the parameters that appear in recently derived mechanism-enabled population balance models (ME-PBM) of nanoparticle growth.

^{[2]}In this paper we use Bayesian inversion to estimate parameters from scatterometry measurements of a silicon line grating and determine the associated uncertainties.

^{[3]}

## Analytical Bayesian Inversion

We demonstrate both methods in an analytical Bayesian inversion of Greenhouse Gases Observing Satellite (GOSAT) methane data with augmented information content over North America in July 2009.^{[1]}Here we constrain emission estimates with the help of satellite observations of carbon monoxide5, an analytical Bayesian inversion6 and observed ratios between emitted carbon dioxide and carbon monoxide7.

^{[2]}An analytical Bayesian inversion model was used to optimize existing emission inventories using long-term, multi-site PM2.

^{[3]}

## Variational Bayesian Inversion

Here, using a variational Bayesian inversion framework and the 3D chemical transport model LMDz, combined with 10 different OH fields derived from chemistry–climate models (Chemistry–Climate Model Initiative, or CCMI, experiment), we evaluate the influence of OH burden, spatial distribution, and temporal variations on the global and regional CH4 budget.^{[1]}1 Journal of Geophysical Research – Solid Earth Supporting Information for Rapid Discriminative Variational Bayesian Inversion of Geophysical Data for the Spatial Distribution of Geological Properties M.

^{[2]}

## Joint Bayesian Inversion

The speed, accuracy and scalability of our open source deep learning models pave the way for extensions of these emulators to generic source mechanisms and application to joint Bayesian inversion of moment tensor components and source location using full waveforms.^{[1]}The speed, accuracy and scalability of our open source deep learning models pave the way for extensions of these emulators to generic source mechanisms and application to joint Bayesian inversion of moment tensor components and source location using full waveforms.

^{[2]}

## Linearized Bayesian Inversion

We present a linearized Bayesian inversion that works in space domain and addresses many issues of previous depth determination approaches.^{[1]}Implementing linearized Bayesian inversion directly in the depth domain using nonstationary wavelets is a convenient new approach that takes advantage of superior computational speed and uncertainty quantification without compromising the accurate spatial location that depth imaging provides.

^{[2]}

## bayesian inversion framework

Using the frequency-dependent viscoelastic impedance equation and Bayesian inversion framework, the objective function of frequency-dependent elastic impedance inversion can be established to realize the frequency-dependent impedance inversion at different angles.^{[1]}In this work an adopted version of a Bayesian inversion framework [1] will be presented.

^{[2]}We apply those three methods to sulfur hexafluoride (SF

_{6}) and use the Bayesian inversion framework FLEXINVERT for the inverse modeling and the Lagrangian particle dispersion model FLEXPART to calculate the source-receptor-relationship used in the inversion.

^{[3]}Here we develop a Bayesian inversion framework that uses Interferometric Synthetic Aperture Radar (InSAR) surface deformation data to infer the laterally heterogeneous permeability of a transient linear poroelastic model of a confined GW aquifer.

^{[4]}To overcome this problem, a nonstationary UTT image reconstruction method is proposed based on the Bayesian inversion framework, where the nonstationary UTT inverse problem is formulated as a state estimation problem using a pair of state evolution and observation update equations.

^{[5]}This method combines a Lagrangian transport model with a Bayesian inversion framework to estimate surface emissions and their uncertainties, together with determining the concentrations of methane in the air flowing into the city.

^{[6]}In this work, Bayesian inversion framework is tested on synthetic microwave tomography data at a single frequency in X-band (8 GHz to 12 GHz).

^{[7]}In this work, a multistatic uniform diffraction tomography (MUDT) method, that was proposed by the authors as a new qualitative imaging method just recently, is combined with the quantitative Bayesian inversion framework.

^{[8]}Here we investigate the effects of the COVID-19 lockdowns in Europe on ambient black carbon (BC), which affects climate and damages health, using in situ observations from 17 European stations in a Bayesian inversion framework.

^{[9]}Objective: To develop a Bayesian inversion framework on longitudinal chest CT scans which can perform efficient multi-class classification of lung cancer.

^{[10]}In this work, a Bayesian inversion framework for hydraulic phase-field transversely isotropic and orthotropy anisotropic fracture is proposed.

^{[11]}This method combines a Lagrangian transport model with a Bayesian inversion framework to estimate surface emissions and their uncertainties, together with determining the concentrations of methane in the air flowing into the city.

^{[12]}In this work, we develop a Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters.

^{[13]}Methane emissions are estimated using a combination of Lagrangian transport model with a Bayesian inversion framework.

^{[14]}We have used a Bayesian inversion framework with l-BGFS algorithm.

^{[15]}Here, using a variational Bayesian inversion framework and the 3D chemical transport model LMDz, combined with 10 different OH fields derived from chemistry–climate models (Chemistry–Climate Model Initiative, or CCMI, experiment), we evaluate the influence of OH burden, spatial distribution, and temporal variations on the global and regional CH4 budget.

^{[16]}In this study, we use a Bayesian inversion framework to compare the estimation uncertainty between broadband and narrowband echo data for biological model parameters, such as organism length, tilt angle, numerical density and aggregation composition.

^{[17]}In this study, we develop an optimal moving window nonlinear Bayesian inversion framework to use the Soil Canopy Observation Photochemistry and Energy fluxes (SCOPE) model for constraining Vcmax , BB slope and LAI with observations of coupled carbon and energy fluxes and spectral reflectance from satellites.

^{[18]}In this study, we develop an optimal moving window non-linear Bayesian inversion framework to use the Soil Canopy Observation Photochemistry and Energy fluxes (SCOPE) model for [.

^{[19]}

## bayesian inversion method

Aiming at the problem of nonlinear inversion of seabed acoustic parameters under the semi-infinite elastic seafloor, the Bayesian inversion method is adopted, and the complex sound pressure at various distance points received by the hydrophone is used as the research object for inversion.^{[1]}Furthermore, a Bayesian inversion method is developed that uses the amplitude variation with angle and azimuth (AVAZ) of the seismic data.

^{[2]}Under Born approximation (BA), this problem is cast into a linear inversion one which is then addressed by means of a hierarchical Bayesian inversion method.

^{[3]}The reversible jump Markov Chain Monte Carlo algorithm is a trans-dimensional Bayesian inversion method, which can not only obtain the best inversion solution, but also provide the uncertainty information of inversion parameters, so as to effectively evaluate the reliability of inversion results.

^{[4]}An observing system simulation experiment (OSSE) is conducted to identify potential locations for making surface ocean pCO2 measurements in the Indian Ocean using the Bayesian Inversion method.

^{[5]}An adjoint Bayesian inversion method was developed to invert for the depth-averaged aquifer petrophysical properties, based on the monitored pressure signals.

^{[6]}We have developed a Bayesian inversion method that integrates the electrical resistivity distribution from MT surveys with borehole methylene blue (MeB) data, an indicator of conductive clay content.

^{[7]}Our results highlight the diversity of dike emplacement histories within the Columbia River Flood Basalt province and the power of Bayesian inversion methods for quantifying parameter trade-offs and uncertainty in thermal models.

^{[8]}We then predicted the risk of colorectal neoplasms on the basis of the corresponding percentile values by using accelerated failure time model with Bayesian inversion method.

^{[9]}Optimal network design, for ground-based monitoring of atmospheric mole fractions of carbon dioxide (CO2) over India to better constrain the Indian terrestrial surface fluxes, is proposed here using a Lagrangian Particle Dispersion Model FLEXPART and Bayesian inversion methods.

^{[10]}The ability to produce meaningful non-linear uncertainty estimates on both conductivity as well as related parameters is a strong motivator for the use of Bayesian inversion methods when computationally feasible.

^{[11]}The Bayesian inversion method is a stochastic approach based on the Bayesian theory.

^{[12]}Furthermore, we apply the Bayesian inversion method to extract the parameter of interest appeared in the analytical formulae of impedance.

^{[13]}We also applied a state‐of‐the‐art Bayesian inversion method to improve the uncertainty estimation of the model parameters.

^{[14]}

## bayesian inversion approach

To solve this problem, this study proposes a new method to estimate the coseismic fault model and model uncertainties in real time based on the Bayesian inversion approach using the Markov Chain Monte Carlo (MCMC) method.^{[1]}In this study, the atmospheric Bayesian inversion approach that couples six in-situ continuous CO2 monitoring stations with the WRF-Chem transport model at 1-km horizontal resolutions has been used to quantify the impacts of lockdown on CO

_{2}emissions for the Paris megacity.

^{[2]}In this study, we use a multi-species atmospheric Bayesian inversion approach to attribute satellite-observed atmospheric CO variations to its sources and sinks in order to achieve a full closure of the global CO budget during 2000–2017.

^{[3]}MCP – which is a Bayesian Inversion approach – was originally developed for predictive uncertainty estimates of water level and discharge to support real-time flood forecasting.

^{[4]}In this study, we use a multi-species atmospheric Bayesian inversion approach to attribute satellite-observed atmospheric CO variations to its sources and sinks in order to achieve a full closure of the global CO budget during 2000–2017.

^{[5]}The Bayesian inversion approach is highly flexible, provides uncertainty quantification, and enables the explicit incorporation of prior knowledge into the inversion process.

^{[6]}A statistical Bayesian inversion approach was used to determine posterior distributions with a particular emphasis on their uncertainty quantification.

^{[7]}In the second part of the work, we present a Bayesian inversion approach for saturation-pressure 4D inversion in which we adopt the new formulation of the rock physics approximation.

^{[8]}Our goal is to demonstrate an azimuthally anisotropic elastic impedance parameterisation and Bayesian inversion approach to estimate orthorhombic anisotropy in a fractured reservoir.

^{[9]}

## bayesian inversion scheme

In this context we propose the use of a greedy reduced basis strategy within a probabilistic Bayesian inversion scheme (MCMC) that makes feasible accounting for the fully dynamic topography model within the inversion.

^{[1]}An alternative full velocity spectrum misfit function and a Bayesian inversion scheme are employed to retrieve tunnel lining structure.

^{[2]}We present a Bayesian inversion scheme to extract multiple bed boundaries from extra-deep directional logging-while-drilling (LWD) resistivity measurements (EDDRM).

^{[3]}Fundamental mode Rayleigh wave phase velocity dispersion data extracted from long-term cross-correlation of ambient noise data are inverted using a transdimensional, hierarchical, Bayesian inversion scheme to produce phase velocity maps in the period range 2–30 s.

^{[4]}For this, we use a Bayesian inversion scheme in which the misfit function is constructed by comparing four geometrical parameters between the natural and the simulated delta: the volume of sediments stored in the delta, the surface slope, the initial and the final shelf lengths.

^{[5]}To account for the strong nonlinearity of the inverse problem and improve inversion efficiency, a Bayesian inversion scheme is formulated using a parallel-tempering Markov chain Monte Carlo approach.

^{[6]}We use a transdimensional Bayesian inversion scheme constrained by geodetic surface displacements and regularized using von Karman correlation.

^{[7]}

## bayesian inversion model

For this problem, we improve the classical Bayesian inversion model by taking into account underestimated uncertainty on reported output observations, which is a frequently encountered issue in practice.^{[1]}The strategy parameterizes the Bayesian inversion model space in terms of sparse, hydrologic-process-tuned bases, leading to dimensionality reduction while accounting for the physics of the target hydrologic process.

^{[2]}The strategy parameterizes the Bayesian inversion model space in terms of sparse, hydrologic-process-tuned bases, leading to dimensionality reduction while accounting for the physics of the target hydrologic process.

^{[3]}An analytical Bayesian inversion model was used to optimize existing emission inventories using long-term, multi-site PM2.

^{[4]}

## bayesian inversion algorithm

The shear-wave velocity and thickness of the sedimentary layers at the investigated slopes are inferred using a transdimensional Bayesian inversion algorithm.^{[1]}We adapt an existing open-source Bayesian inversion algorithm, which uses independent depth constraints, to output posterior distributions of resistivity and pore fluid salinity with depth.

^{[2]}The input of a Bayesian inversion algorithm is received different normal mode impulse signals, which are separated and extracted with a warping transformation from received broadband impulse signals.

^{[3]}

## bayesian inversion procedure

Their construction as linear unital maps are obtained via a categorical Bayesian inversion procedure.^{[1]}A recently introduced transdimensional Bayesian inversion procedure is applied for both tomographical methods, which adjusts the fracture positions, orientations, and numbers based on given geometrical fracture statistics.

^{[2]}

## bayesian inversion result

The coloured and Bayesian inversion results were generally consistent with well-log observations at the reservoir level and conformed to interpreted horizons.^{[1]}We apply the method to Schiehallion field data and go on to compare the results to Bayesian inversion results.

^{[2]}

## bayesian inversion proces

The Bayesian inversion process was conducted with Metropolis-Hastings algorithm and maximum a posteriori.^{[1]}We constrained the models based on the results of a Bayesian inversion process using Rayleigh wave dispersion data, which were measured from ambient noise cross-correlations between stations in the southern Korean Peninsula and northeast China.

^{[2]}

## bayesian inversion theory

The dispersion curves extracted from the measured data is used as the input signal of Bayesian inversion theory, and the replica is calculated by dispersion formula, which is composed of bottom reflection phase shift parameter, depth, propagation range and average sound velocity in water.^{[1]}Finally, based on the ingeniously developed nonlinear Bayesian inversion theory, the seafloor shear wave velocity profile in the southern Yellow Sea of China is inverted by employing multi-order Scholte wave dispersion curves.

^{[2]}